Fractional Quantum Hall Phase Transitions and Four-Flux States in Graphene

Benjamin E. Feldman,1 Andrei J. Levin,1 Benjamin Krauss,2 Dmitry A. Abanin,1,3

Bertrand I. Halperin,1 Jurgen H. Smet,2 and Amir Yacoby1,*1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

2Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 6B9, Canada

(Received 7 May 2013; published 16 August 2013)

Graphene and its multilayers have attracted considerable interest because their fourfold spin and valley

degeneracy enables a rich variety of broken-symmetry states arising from electron-electron interactions, and

raises the prospect of controlled phase transitions among them. Here we report local electronic compressi-

bility measurements of ultraclean suspended graphene that reveal a multitude of fractional quantum Hall

states surrounding filling factors � ¼ �1=2 and�1=4. Several of these states exhibit phase transitions that

indicate abrupt changes in the underlying order, and we observe many additional oscillations in compressi-

bility as � approaches�1=2, suggesting further changes in spin and/or valley polarization. We use a simple

model based on crossing Landau levels of composite fermions with different internal degrees of freedom to

explainmany qualitative features of the experimental data. Our results add to the diverse array ofmany-body

states observed in graphene and demonstrate substantial control over their order parameters.

DOI: 10.1103/PhysRevLett.111.076802 PACS numbers: 73.22.Pr, 73.43.�f

When a two-dimensional electron gas is subject to aperpendicular magnetic field B, the electronic spectrumforms a sequence of Landau levels (LLs). Generally, thisgives rise to incompressible quantized Hall states at integervalues of the filling factor � ¼ nh=eB, where n is thecarrier density, h is Planck’s constant, and e is the electroncharge. In very clean samples at high magnetic field,Coulomb interactions become important and produce addi-tional quantized Hall states at certain fractional fillingfactors [1–4]. These fractional quantized Hall (FQH) statescan be understood in terms of composite fermions (CFs),which may be described as an electron bound to an evennumber m of magnetic flux quanta. CFs experiencea reduced effective magnetic field, and FQH states at� ¼ p=ðmp� 1Þ are understood to arise when an integernumber p of mCF LLs are occupied. In CF theory, FQHstates of electrons are therefore interpreted as the integerquantized Hall effect of these new composite particles [4].

Like electrons, CFs can have internal quantum numberssuch as spin or valley index (isospin). When more than oneCF LL is occupied, ground states with different polariza-tions of these degrees of freedom are possible at a givenfilling factor, and transitions between different phases mayoccur when system parameters are varied. Phase transitionsbetween FQH states with differing spin polarization havebeen observed in GaAs by tuning the magnitude of themagnetic field [5–8], its direction [9–14], or the appliedpressure [15]. In AlAs 2DEGs, strain has been used toinduce phase transitions between valley-polarized andunpolarized states [16–18].

In graphene, the electronic Hamiltonian has anapproximate SU(4) symmetry arising from the spin andvalley degrees of freedom. This symmetry is weakly brokendue to the Zeeman effect and electron-electron scattering

between valleys, which may be enhanced by (or competewith) effects of the dominant Coulomb interactions.Electron-electron interactions were recently shown to pro-duce surprising patterns of symmetry breaking and phasetransitions in the integer quantum Hall regime [19–23].Theoretical proposals suggest that the strengths of FQHstates can also be tuned in monolayer and bilayer graphene,and that transitions between different ordered phases arepossible [24–26]. However, despite the observation ofrobust FQH states in graphene [27–32], their rich phasediagram has yet to be fully explored.Here we report local electronic compressibility mea-

surements of suspended graphene, performed using a scan-ning single-electron transistor (SET) [33,34]. A schematicof the measurement setup [35] is shown in Fig. 1(a).Modulating the carrier density with a back gate and moni-toring the resulting change in SET current allows us tomeasure both the local chemical potential � and the localinverse compressibility of the graphene flake with a spatialresolution of about 100 nm. The inverse compressibility��1 is properly defined as n2d�=dn, but hereafter we dropthe prefactor and use the term to mean d�=dn. The datapresented below were taken at one location, but similarbehavior was observed at multiple positions [35].Figure 1(b) shows the inverse compressibility as a func-

tion of filling factor and magnetic field. FQH states appearas vertical incompressible peaks at � ¼ �1=3, �2=3,�2=5, �3=5, �3=7, �4=7, �4=9, �5=9, and �5=11,consistent with the standardCF sequence observed for j�j<1 in previous measurements [32]. Surprisingly, every FQHstate except � ¼ �1=3 exhibits a narrow magnetic fieldrange over which the incompressible behavior is stronglysuppressed. The critical field at which this occurs increaseswith filling fraction denominator, and the suppression is

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associated with regions of sharply negative compressib-ility that cross each FQH state, often generating two coex-isting incompressible peaks at slightly different fillingfactors over a small range in magnetic field [Fig. 1(c)].Interestingly, the negative compressibility has an especiallylarge amplitude that is similar (but opposite in sign) to theincompressible peaks of the actual FQH states.

The interruptions in each incompressible peak suggestphase transitions in which the spin and/or valley polariza-tion of electrons changes abruptly. The behavior is similarto that observed in GaAs, where transport measurementsshowed FQH states splitting into doublets near phase tran-sitions [6,7,11,14]. However, no dramatic features of nega-tive compressibility were present in GaAs [8], and theinverse compressibility did not display a strong asymmetrybetween filling factors just above and below the FQH states[8,36].

Several less prominent modulations in compressibilitythat occur close to � ¼ �1=2 are also visible in Fig. 1(b).We emphasize that they are not caused by localized states,which occur near the strongest FQH states such as � ¼�2=3, but not around high-denominator states, such as� ¼ �4=7 [35]. Further insight can be gained by plottingthe inverse compressibility as a function of p rather than �[Figs. 1(d) and 1(e)]. This more clearly illustrates thebehavior near � ¼ �1=2 and reveals oscillations in inversecompressibility that persist to values of p as large as 20 andmagnetic fields as low as a few Tesla. The behavior cannotbe explained by Shubnikov–de Haas oscillations of CFs,because variations in compressibility occur even at

constant filling factor. The oscillations become strongerand more vertical as the magnetic field is increased, sug-gesting that they are associated with developing FQHstates. Moreover, they seem to extend from the negativecompressibility features of the phase transitions, suggest-ing that they result from changes in spin and/or valleypolarization as magnetic field and filling factor are varied.Signatures of phase transitions have previously been

observed in compressibility measurements only at � ¼2=3 in GaAs [8], although optical and transport studiesof GaAs and AlAs have revealed evidence of changes inspin or isospin polarization for filling fractions with largerdenominators [9,37]. We observe clear phase transitions upto � ¼ �5=9 and �5=11, and additional compressibilityoscillations are apparent much closer to � ¼ �1=2.Similar oscillations have not been reported in GaAs; theirexistence in graphene suggests a rich array of orderedelectronic states and hints at a delicate energetic competi-tion among them.To gain further insight, we introduce a simple model to

describe CFs with internal degrees of freedom [38–41] (see[35] for details). Because of graphene’s peculiar bandstructure, the lowest LL is already half full at � ¼ 0, andexperiments suggest that the � ¼ 0 state has no net spinpolarization [19]. For 0> �>�1, we assume that theground state is obtained by putting holes in the � ¼ 0 state,which we convert to CFs by attaching two flux quanta toeach hole. The CFs have two possible spin states (�) andwe consider many-body states where there may be differ-ent particle densities for the two spins. Single-particle

ν

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FIG. 1 (color). (a) Schematic of the measurement setup. (b) Inverse compressibility d�=dn as a function of filling factor � andmagnetic field B. (c) Finer measurement around � ¼ �4=7. Panels (b) and (c) have identical color scales. (d) and (e) d�=dn as afunction of B and composite fermion Landau level (CF LL) index p. Panels (d) and (e) have identical color scales. Principal FQH statesare marked by black lines and are labeled. The dashed lines in panel (b) mark where higher-denominator FQH states in the standard CFsequence would be expected to occur.

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energies of the two spin states will be split by an amountproportional to B due to the Zeeman effect, which favors aspin-polarized state. However, the SU(4) invariant part ofthe Coulomb interaction will typically favor states withmore equal occupation of spins [39]. Because the Coulomb

interaction energies scale as B1=2, then for fixed �, varyingthe magnetic field will change the relative importance ofthe two terms, which suggests that the experimentallyobserved phase transitions may be caused by changes inspin polarization, as in GaAs.

Our model applies most directly to the situation whereall electrons in the ground state of � ¼ 0 have the samevalley configuration, as in the Kekule or the charge-density-wave states [42]. The antiferromagnetic state ismore complicated because the constituent electron statesdiffer in valley index as well as spin, but we expect thatresults for this case should be at least qualitatively similarto the case we consider [43]. Future studies in which atilted magnetic field is applied to the sample may helpdetermine the spin and valley ordering of the FQH states.

Within our model, effects of the SU(4) invariantCoulomb interaction are modeled by a sum of the ‘‘kineticenergies’’ of the occupied states in the CF LLs, which scale

as B1=2 for fixed orbital index N�. A schematic diagram ofCF LL energies E�

N� and their scaling with magnetic field is

shown in Fig. 2(a). At certain critical magnetic fields, CFLLs with different spin and orbital degrees of freedomcross, leading to phase transitions.

Based on this model, we have numerically simulated theinverse compressibility. In our simulation, we broadenthe CF LLs by a fixed amount of disorder �n and calculatethe occupation of each CF LL, which ultimately yields theinverse compressibility as a function of density andmagnetic field. The results, which assume either a smallamount of disorder or more realistic density fluctuationsbased on the widths of the FQH peaks, are shown inFigs. 2(b) and 2(c), respectively.

The simulations in Fig. 2 share many characteristicswith the experimental data. Most striking are the breaksin the incompressible peaks of FQH states with p � 2. Inaddition, the simulations show regions of negative com-pressibility that cross from one side of the FQH state to theother as the phase transition occurs. This is qualitativelysimilar to the behavior that we observe, although theexperimental features are much narrower. Finally, theoscillations in inverse compressibility become less robustand start to curve at low magnetic field and high p, similarto the experimental data. The values used for parameters inthe simulation agree well with expectations based on otherexperimental metrics. By matching the simulation to theexperimental critical fields and assuming Zeeman splitting

with a g factor of 2, we extract an effective mass m� ¼0:18með�B½T�Þ1=2, the same order of magnitude as for CFsin GaAs [5]. In addition, the density fluctuations �n ¼1:5� 108 cm�2 assumed in Fig. 2(c) are comparable to thewidths of the FQH states we observe. Given the simplicity

of the model, the agreement with experiment is remark-able, suggesting that it provides a basic framework tounderstand the underlying physics.The critical fields of the phase transitions vary slightly

with position, and a much smaller critical field at � ¼ 2=3was observed before the final current annealing step [35].The change after current annealing suggests that disorder isrelevant, but the exact mechanism is not clear. Disorderthat breaks valley symmetry could preferentially supportone FQH phase over the other if the � ¼ 0 state is a cantedantiferromagnet. It is also possible that the level of disorderaffects the dielectric constant. The origin of the spatialdependence merits further study.Integrating the inverse compressibility with respect to

carrier density allows us to extract the steps in chemicalpotential ��� of each FQH state; multiplying ��� by thequasiparticle charge then yields the corresponding energygaps. Figures 3(a) and 3(b) show inverse compressibilityand chemical potential, respectively, as a function of fillingfactor at 11.9 T. In Figs. 3(c) and 3(d), we plot ��� as afunction of magnetic field. The complex nonmonotonicbehavior of the energy gaps exhibited by several FQHstates [Fig. 3(d)] is similar to the behavior in GaAs near

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and plotted against B1=2. Crossings (black circle) between spin-up and -down CF LLs (colored arrows) correspond to phasetransitions. (b) and (c) Numerical simulations of d�=dn as afunction of B and p assuming either minimal charge inhomoge-neity (b), or more realistic density fluctuations (c). Black ovalscorrespond to the black circle in panel (a). Both panels use thesame color scale.

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phase transitions [13]. This behavior becomes increasinglypronounced and the field range over which incompressiblebehavior is suppressed widens as filling factor denominatorincreases. A similar pattern occurs in the simulations ofFig. 2, and it likely results from the increasing effects ofdensity fluctuations on CF LL width as p increases.

The step in chemical potential at � ¼ �1=3 scaleslinearly with B over the entire field range that we study.This behavior is consistent with prior studies [32], but thelinearity is surprising because interaction-driven states

typically scale as B1=2. The behavior also contradicts the

B1=2 dependence expected from our model, although wenote that the model does not include interactions amongCFs, LL mixing, finite temperature effects, or the possi-bility of other excitations such as Skyrmions. Linear scal-ing with magnetic field at � ¼ 1=3 has been theoreticallypredicted to arise from spin-flip excitations over anintermediate field range [44].

In addition to the phase transitions discussed above, theexceptional sample quality reveals several FQH statesbelonging to the 4CF sequence � ¼ p=ð4p� 1Þ and itsanalogue around � ¼ �1. We observe incompressiblebehavior at � ¼ �1=5, �2=7, �2=9, �3=11, �5=7 and�6=5 [Figs. 4(a)–4(c)]. An additional weak incompress-ible peak occurs between � ¼ �9=7 and �14=11, but theexperimental uncertainty in filling factor prevents a moreprecise assignment [35]. No other 4CF states are visible;FQH states at � ¼ �4=5, �9=5, and �12=7 are conspic-uously absent, despite the robust appearance of their coun-terparts near � ¼ 0. This may reflect interesting patterns ofsymmetry breaking in the lowest LL [35,45], but could alsobe caused by differing degrees of disorder at differentfilling factors, or by competition with other quantum Hallstates, particularly near � ¼ �2.

The extracted steps in chemical potential for several4CF FQH states are plotted as a function of magnetic field

in Fig. 4(d). The fluctuations caused by localized statesnear � ¼ �1=5 and �2=9 prevent an accurate determina-tion of ��� for these states, but all other states except for� ¼ �5=7 scale approximately linearly with magneticfield. Further study is required to determine whether thenonmonotonic behavior of ���5=7 reflects a phase tran-

sition or whether the state is simply competing with � ¼�2=3. Regardless, the appearance of 4CF states and phasetransitions represents an important advance in samplequality that enables further study of and control over thedelicate many-body states arising from interacting Diracfermions in graphene.We would like to thank M.T. Allen for help with current

annealing.We also acknowledge useful discussions withM.Kharitonov, J. K. Jain, L. S. Levitov, and S.H. Simon. Thiswork is supported by the U.S. Department of Energy, Officeof Basic Energy Sciences, Division of Materials Sciencesand Engineering under Award No. DE-SC0001819. J. H. S.and B.K. acknowledge financial support from the DFGgraphene priority programme. B.K. acknowledges financialsupport from the Bayer Science and Education Foundation.This workwas performed in part at the Center for NanoscaleSystems at Harvard University, a member of the NationalNanotechnology Infrastructure Network, which is sup-ported by the National Science Foundation under AwardNo. ECS-0335765.

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1

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4/94/7 5/93/7

FIG. 3 (color). d�=dn (a) and chemical potential � relative toits value at � ¼ �1=2 (b) as a function of � at B ¼ 11:9 T. (c)and (d) Steps in chemical potential ��� [green labels in panel(b)] of FQH states as a function of B at measured multiples of� ¼ 1=3 and 1=5 (c), and � ¼ 1=7 and 1=9 (d).

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